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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Ma Shohei
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (H201)
- Group
- -
- Course number
- MTH.E432
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 3Q
- Syllabus updated
- 2016/4/27
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The fundamental concepts are:
Riemann-Roch theorem, Serre duality, canonical model, Hodge decomposition, Torelli theorem
Solve the Exercises presented in the lecture
Student learning outcomes
We will study the basic theory of complex algebraic curves. There are two main topics: (1) the Hodge decomposition of the cohomology, which leads to the Jacobian, the Abel-Jacobi map and the Torelli theorem; and (2) the canonical model which studied using the cohomology of line bundles. These two topics are closely related by the Gauss map of Abel-Jacobi image.
Keywords
algebraic curves, Riemann surfaces
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | algebraic curves | Details will be provided during each class session |
| Class 2 | line bundles | Details will be provided during each class session |
| Class 3 | cohomology | Details will be provided during each class session |
| Class 4 | Riemann-Roch formula | Details will be provided during each class session |
| Class 5 | Serre duality | Details will be provided during each class session |
| Class 6 | canonical model | Details will be provided during each class session |
| Class 7 | Clifford's theorem | Details will be provided during each class session |
| Class 8 | Green conjecture | Details will be provided during each class session |
| Class 9 | Jacobi variety | Details will be provided during each class session |
| Class 10 | Abel-Jacobi map | Details will be provided during each class session |
| Class 11 | Torelli theorem | Details will be provided during each class session |
| Class 12 | theta divisor | Details will be provided during each class session |
| Class 13 | moduli space | Details will be provided during each class session |
| Class 14 | canonical divisor of Mg | Details will be provided during each class session |
| Class 15 | Harris-Mumford theorem | Details will be provided during each class session |
Textbook(s)
None in particular
Reference books, course materials, etc.
E.Arbarello, M.Cornalba, P.Griffiths, J.,Harris, `Geometry of Algebraic Curves I' Springer.
R.Narashimhan, `Compact Riemann surfaces'
J.Harris, I.Morrison, `Moduli of Curves' Springer
Assessment criteria and methods
by Report. Details will be provided in the class.
Related courses
- ZUA.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None in particular
